Relative Gromov-Witten invariants, which count curves with specified orders of tangency with a smooth divisor, have proven to be an extremely useful tool in Gromov-Witten
theory: the gluing formula allows one to compute GW invariants of varieties by degenerating them to normal crossing unions of two varieties and then computing relative GW invariants on each of these varieties. Siebert and I propose a generalization of relative GW invariants which will allow tangency conditions with much more general divisors, and allow much more complicated degenerations.

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